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2007 Fiscal Year Final Research Report Summary

Solution of Broue's conjecture in representation theory of finite groups

Research Project

Project/Area Number 17540010
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionChiba University

Principal Investigator

KOSHITANI Shigeo  Chiba University, Graduate School of Science, Professor (30125926)

Co-Investigator(Kenkyū-buntansha) NOZAWA Sohei  Chiba University, Graduate School of Science, Professor (20092083)
KITAZUME Masaaki  Chiba University, Graduate School of Science, Professor (60204898)
NISHIDA Koji  Chiba University, Graduate School of Science, Associate Professor (60228187)
MATSUDA Shigeki  Chiba University, Graduate School of Science, Associate Professor (90272301)
OTUBO Noriaki  Chiba University, Graduate School of Science, Assistant Professor (60332566)
Project Period (FY) 2005 – 2007
Keywordsfinite group / representation theory / modular representation theory / block / abelian defect group / Broue's conjecture / Morita equivalence / irreducible character
Research Abstract

The head investigator Shigeo Koshitani has gotten the following results in the last three yeas.
Firstly, Koshitani worked with Morton E.Harris (University of Illinois at Chicago, and we proved that there is a Morita equivalence between two block algebras, say A and B such that A and B are corresponding via the Glauberman corresondence and moreover, the Morita equivalence is described explicitly by an (A,B)-bimodule M which induces the original Glauberman correspondence at the level of ordinary characters. This result has been published in the Journal of Algebra (Elsevier).
Secondly, Koshitani has been working with Naoko Kunugi and Katsushi Waki together, in order to solved Broue's abelian defect group conjecture for the fourth Janko simple group J4. This group J4 is relatively quite large in the twenty-six sporadic simple groups. The three people above proved that Broue's abelian defect group conjecture is true for the group J4 for all prime numbers. This result was published in the Jour … More nal of Pure and Applied Algebra (Elsevier). Koshitani has been working on also another kind of Broue's conjecture. Namely, he has been looking at blocks of finite groups whose defect groups are not abelian but metacyclic groups. He with other two co-researchers has gotten some results on the subject as well.
The head investigator Shigeo Koshiani announced and gave talks on the above results in many places. Such as at the Mathematical Institute of Oberwolfach, Germany (2006), the Mathematics Luminy Institute, France (2007),University of Chicago, Mathematical Institute of the University of Oxford, University of Aberdeen in the UK, University of Leeds in the UK, University of Jena, Germany, University of Braunsweich Technology, Germany, National University of Ireland in Maynooth, Kyoto University, and so on.
On the other hand, Naoko Kunugi, who has been one of the investigators of this project also gave several invited talks on Broue's abelian defect group conjecture. For instance, she gave two talks in the Symposium on algebraic groups and quantum groups held in May 2006 near Nagoya, Japan, and also she gave an invited special talk at the annual meeting orthorized by the Mathematical Society of Japan in September 2006 in Osaka City University. Less

  • Research Products

    (8 results)

All 2008 2006

All Journal Article (6 results) (of which Peer Reviewed: 3 results) Presentation (2 results)

  • [Journal Article] Broue's abelian defect group conjecture Rolds for the Janko simple group J42008

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Pure and Applied Algebra 212

      Pages: 1438-1456

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Broue's abelian defect group conjecture holds for the Janko simple group J42008

    • Author(s)
      Shigeo Koshitani, Naoko Kunugi and Katsushi Waki
    • Journal Title

      Journal of Pure and Applied Algebra 212

      Pages: 1438-1456

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An extension of watanabe's theorem for the Isaacs-Horimoto- watanabe corresponding blooks2006

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Algebra 296

      Pages: 96-109

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] On Burch's ineguality and a reduction system of a filtration2006

    • Author(s)
      Koji Nishida
    • Journal Title

      Proceedings of the American Mathematical society 134

      Pages: 3437-3444

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] An extension of Watanabe's theorem for the Isaacs-Horimoto-Watanabe corresponding blocks2006

    • Author(s)
      Morton E. Harris and Shigeo Koshitani
    • Journal Title

      Journal of Algebra 296

      Pages: 96-109

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On Burch's inequality and a reduction system of a filtration2006

    • Author(s)
      Y. Kinoshita, H. Nishida, Y. Yamanaka and A. Yoneda
    • Journal Title

      Proceedings of the American Mathematical Society

      Pages: 3437-3444

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Thvial source modules in blocks with cyclic defect groups2006

    • Author(s)
      Shigeo Kashitani
    • Organizer
      ドイツ・オーバーヴォルファッハ研究所有限群の表現
    • Place of Presentation
      ドイツ・オーバーヴォルファッハ数学研究所
    • Year and Date
      2006-03-29
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Trivial source modules in blocks with cyclic defect groups2006

    • Author(s)
      Shigeo Koshiani
    • Organizer
      Meeting "Representations of Finite Groups"
    • Place of Presentation
      Oberwolfach Mathematical Institute
    • Year and Date
      2006-03-29
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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